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Expansion of random boundary excitations for elliptic PDEs. (English) Zbl 1158.60027
The author constructs exact proper orthogonal decomposition for some classical boundary value problems for a disc, the ball, and a half-plane, with Dirichlet and Neumann boundary functions which are white noise or homogeneous (2\(\pi\)-periodic) random processes. For example the author considers the Dirichlet boundary value problem for the Laplace equation
\[ \Delta u(x) =0, \quad x \in D, \quad u(y) =g(y),\quad y \in \Gamma = \partial D, \tag{1} \]
where the domain \(D\) is a disc centered at 0; \(g(y)\) is a zero mean Gaussian random field defined by its correlation function \(B_g(y_1, y_2).\)
The following theorem is obtained. Theorem 1. The solution of the Dirichlet problem (1) in a disc \(K(x_0, R)\) with the white noise boundary function \(G(y)\) is an inhomogeneous 2D Gaussian random field uniquely defined by the its correlation function
\[ \langle u(r_1, \theta_1) u(r_2, \theta_2)\rangle = B_u (\rho_1, \theta_1; \rho_2, \theta_2) = {1 \over 2\pi} {1- \rho_1^2 \rho_2^2 \over 1- 2\rho_1 \rho_2 \cos (\theta_2 - \theta_1) + \rho_1^2 \rho_2^2 } \]
which is harmonic, and it depends only on angular difference \(\theta_2 -\theta_1\) and the product of radial coordinates \(\rho_1 \rho_2 / R^2.\) The random fields \(u(r, \theta)\) is thus homogeneous with respect to the angular coordinate \(\theta,\) and its partial discrete spectral density has form \(f_\theta (0) =1/ 2\pi,\) \(f_\theta (k) = (\rho_1 \rho_2)^k/ \pi,\) \(k=1,2,\dots\).

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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